MODELLING OF THE RACH CHANNEL IN A REAL ENVIRONMENT FOR A HIGH EFFICIENCY AND STABILITY ON WIRELESS COMMUNICATIONS Aldo Mendez1, David Covarrubias1, and Cesar Vargas2 1
Wireless Communications Group-CICESE Research Centre, km. 107 Carretera Tijuana-Ensenada, 22860, Ensenada-Baja California, México, phone: (52+646) 1750555, fax: (52+646) 1750554, e-mail: [amendez, dacoro]@cicese.mx 2 ITESM-CET Campus Monterrey, Av. Eugenio Garza Sada 2501 Sur CETEC 7o Piso TS, 64849, Monterrey-Nuevo León, México, phone: (52+81) 81582081, fax: (52+81) 83597211, e-mail:
[email protected]
Abstract - The purpose of this work is to present realistic behaviour of Slotted Aloha (S-ALOHA), used as Request Access Channel (RACH), with good performance regarding stability, efficiency, delay and capacity of traffic control. We present modelling and simulation of those parameters, which most influence the performance in terms of stability, and efficiency of the MAC technique based on S-Aloha. Considering particularly those aspects that degrade in greater scale the response of the system, such as: retransmission algorithm, capture effect and radio channel effects, we show that throughput can be improved in 70% and can be stabilised. Therefore, if we use S-Aloha as Request Access Channel (RACH), it could be a good alternative for third-generation systems.
The organisation of this work, shows in Section II, the Markov modelling of the MAC technique considering SAloha for the RACH channel. We include in our modelling, the effect of the presence of the radio channel, measured in the terms of the capture effect, and spatial distribution of the mobiles. We suggest, in the same stage of modelling, the use of an algorithm of retransmission adaptable to the conditions of traffic changes. In Section III, the simulation and the numerical analysis of the response of the system are presented. Considering each one of the parameters mentioned, we establish their influence level on the global performance of the system, in terms of efficiency, stability and capacity in traffic management.
Keywords - MAC, radio channel, retransmission algorithm, S-Aloha, stability.
II. MODELLING OF THE RACH CHANNEL
I. INTRODUCTION It is worth emphasising that given the present randomness characteristics in a mobile communication scenario, it is necessary to count with a MAC technique, that can efficiently cope with the possible interference among mobile terminals (MTs) which use simultaneously the RACH channel directed toward the same base station (BS), i.e., the MAC technique should help prevent and solve these problems, as well as optimise the request of the channel in order to have high throughput and low average delay. It is well known that S-Aloha has serious limitations in terms of throughput (0.368 is the maximum), and has stability problems as traffic increases, [1]-[4]. In this work, we carry out a modelling, and a numerical analysis of two fundamental parameters in the performance of S-Aloha: efficiency and stability, obtaining values to achieve all together the best performance possible of this MAC technique. We also show simulation results, interesting operation parameters associated to the communication system, such as: traffic generated, offered traffic, throughput, average delay and the number of mobile terminals in backlog.
0-7803-7589-0/02/$17.00 ©2002 IEEE
The initial stage of the analysis and modelling of the RACH channel is made on a slot by slot basis. Moreover, let ηs(k) denote the number of backlogged mobile terminals at the beginning of the k-th slot, and the process could be represented by a Markov chain modelled by a birth-death process. According to the analysis made in [5], the process of retransmission (RTx) and transmission (Tx) of each mobile terminal is an independent geometric process, in which the probability that i out of the j backlogged mobile terminals program a RTx in a single slot, represents a binomial distribution, with probability that a mobile terminal retransmits a packet ν, and probability that a mobile terminal generates a new packet ϕ. Therefore, the steadystate transition probabilities, p ij = lim Pr (η s (k ) = j | η s (k − 1) = i ) are obtained and the k →∞
transition matrix P is formed [6]. The steady-state probability vector π, whose elements are πj is the solution to M
the finite set of linear equations π=πP, and ∑ π i = 1, [7], i=0
[8]. Through computer simulations, we have obtained the behaviour of the steady-state probabilities considering the
PIMRC 2002
analysis made in [5] for a finite population and the limiting distribution [6]. In Figure 1 we show, as an example, part of the results obtained.
Psucc (i ) = (1 − v )i ⋅ (M − i ) ⋅ ϕ ⋅ (1 − ϕ)M −i −1 i 1 M i + i ⋅ v ⋅ (1 − v ) − ⋅ (1 − ϕ) −
,
(1)
and the throughput S, is expressed as M
S = E [ Psucc ( i )] = ∑ Psucc ( i ) ⋅ π i ,
(2)
i=0
where the vector of probability of the steady state • can be calculated according to what has been discussed in the previous section. B. Modelling the Capture Effect of the RACH Channel The fact that the packet reaches the receiver with different power levels makes possible the capture effect. In other words, the signal with greater intensity can be captured by the receiver. Hence, the probability of mutual destruction of the colliding packets is reduced, resulting in an increase in the efficiency of the system, [10].
a) 1.0
Considering the modelling carried out of the throughput of S-Aloha as a RACH channel in Section II.A, now we add the capture effect. Under this new scheme, the state transition probability can be determined according to [11][13]
0.9
Steady-State Probability - π
0.8 ν=0.05
0.7
ν=0.1
@ ϕ=0.1
ν=0.2
0.6
ν=0.3
0.5 0.4 0.3 0.2 0.1 0.0 0
2
4
6
8
10
12
14
16
18
20
Backlogged Terminals
b) Fig.1. Steady-state probability of the RACH channel SAloha: a) Representation in the limiting distribution. b). Profile of the limiting distribution. Figure 1 shows us that as the probability of retransmission, ν, increases, there exists a high probability of obtaining all the MTs in backlog, which suggests the design of an algorithm that can handle the probabilities of retransmission adaptably and dynamically. If the probability of generation and retransmission are high, all the MTs will be in backlog, and it will be necessary to apply a scheme Deferred First Transmission – DFT [9], with the idea to treat equally all the packets (new and backlogged), facilitating the calculation of the retransmission probability. A. Modelling the Throughput of the RACH Channel For a transmission to be successful, there should only be a single transmission in the slot. This indicates that all of the mobile terminals in backlog are in silence and that only a new mobile terminal transmits, or only one mobile terminal in backlog transmits while there is no new packet generated [10]. So the probability of success when i mobile terminals are in backlog state, is given by
0 , i c i −c i Pcapt (c ) , M −i ν (1 − ν ) ∑ c (1 - ϕ) c =1 M − i j −i +1 ϕ (1 − ϕ)M − j −1 p ij = 1{M 〉 j} 1 j i − + i i k ⋅ ∑ ν (1 − ν )i −k Pcapt (k + j − i + 1) k =0 k + M − i ϕ j −i (1 − ϕ)M − j j − i i i k i −k ⋅ ∑ ν (1 − ν ) 1 − Pcapt (k + j − i ) , k =0 k
[
]
j 〈i - 1 ,
j = i -1 ,
(3)
M ≥ j ≥i.
The probability of success, Psucc(i) in Equation (1), be rewritten to consider the capture effect as M − i M − i r Psucc ( i ) = ∑ ϕ (1 − ϕ)M − i − r M i r r =0 − − i i c i c ⋅ ∑ ν (1 − ν ) − Pcapt (c + r ) . c c =0
(4)
The throughput, S, considering the capture effect, will be given by M
S = E[Psucc (i )] = ∑ Psucc (i ) ⋅ π i . i =0
(5)
According to equations (4) and (5), we need to determine the capture probabilities Pcapt. These probabilities are obtained considering a defined real scenario by a spatial distribution of the mobiles within a uniform type cell, and the presence of the radio channel characterised by Rayleigh fading and shadowing. C. Spatial Distribution of the Mobile Terminals Another point to consider is to find the modelling of spatial distribution that best adjusts to the mobility of the mobile terminals. According to [14] the difference between the performance of S-Aloha when considering uniform and nonuniform spatial distributions (particularly Bell-Shaped) are minimum (about 6%), so given the ease of mathematical treatment we will use in our modelling a uniform type distribution. D. Presence of the Radio Channel
∞∞1
0 00 −
(
1
(A − log u )2
⋅e
where B =
−
log e 2πσ
2σ 2
)
Ry 4 z + u ⋅ z
⋅e
(A − log z )2 2σ 2
∧ 1 ν n = min1, . In this algorithm, all the mobile terminals ∧ n
observe the feedback channel. Thus, they obtain information about the outcome of each slot. In slot k+1, every mobile terminal independently performs an update of his estimate ∧
∧
n and obtains an n k +1 depending on the outcome of the
previous slot [11] est _ i , if slot k was idle, ∧ n k +1 = n + est_s , if slot k had one success (active state), est_c , if slot k had one collision (backlog state),
∧
(7)
est _ i = 2 − e , est _ s = 2 −
e , 1 − pe
(8)
est _ c = 2 .
where pe is the probability of the packet not being accepted by the receiver. With these values, it is possible to calculate in a dynamic form the estimate, Equation (7), obtaining control of the load of the system. Therefore, the objective is to operate with a binary exponential retransmission algorithm [9] that permits us to stabilise the response of SAloha, as it will be seen in Section III.
(6)
⋅ dydudz ,
, A = log W0 +
∧
revealed as n , [1], [18]. Under these circumstances, each backlogged packet is then retransmitted with a probability
where est_i, est_s, est_c are defined as:
The radio signal that is transmitted suffers fading caused by multipath and shadowing caused by obstacles, in which case the power of the signal can be characterised by a log-normal probability density function, [15], [16] with a standard deviation, σ, of 5 dB for an outdoor cellular environment. The fading of the instantaneous envelope of the received signal follows a Rayleigh pdf, [15], [17]. So according to [10], the expression for the probability of capture considering the uniform spatial distribution and the effect of the channel (Rayleigh + Shadowing) is given by
Pcapt = B 2 ⋅ ∫ ∫ ∫ 2 y ⋅
performing, at the beginning of each slot, a load estimation
σ2 , W0 is the mean value 2 log e
of the signal power, σ is the standard deviation, y is the distance between the MT and BS, and R the capture ratio. E. Analysis of Stability of the RACH Channel Another parameter of vital importance in the presence of SAloha, acting in the RACH Channel is the stability. Given the direct relationship between stability and the algorithm of retransmission of the MAC protocol, what we look for is to define an algorithm that can use a dynamic control of the retransmission probabilities; so it improves the capacity of the system in the traffic management. According to the feedback information of a slot (idle, success, and collision), we have modelled the algorithm of retransmission through Markov chains. With this information the algorithm of retransmission has the capacity to carry out certain load control in the channel by
III. NUMERICAL ANALYSIS In this discrete simulation process the packets of each MT are generated according to a Bernoulli process with probabilities of generation of 10-3 to 1. For each probability of generation, 10 000 packets have been transmitted, and the probability of transmitting a new packet is 1, independently of the present value of p (probability of retransmission). We also assume that an MT cannot generate a new packet until the present packet has been transmitted. Additionally, Table 1 shows parameters considered in the simulation Table 1 Parameters of Simulation Parameters
Value
Number of mobile/cellular terminals - M
80
Standard deviation for outdoors - σ
5 dB
Power loss factor - α
4
Signal to interference ratio (SIR)
9 dB
MTs power of transmission
125 mW
In our presentation of results, we will be including, in a gradual manner the described parameters of Section II, and the first parameter to simulate is the throughput. A. Behaviour of the Throughput of S-Aloha Figure 2 shows the response of S-Aloha in our simulation, considering a uniform type spatial distribution of the mobiles and capture effect without considering effect of the radio channel. The parameter R means the capture ratio which we varied from the almost perfect capture (R=2) up to imperfect capture (R=10) [14].
C. Behaviour of the Number of Mobile Terminals in Backlog Mode
0.6
0.5
Uniform Distribution S-Aloha R=2 S-Aloha R=4 S-Aloha R=10 S-Aloha
0.4
According to our simulation and model, and as a consequence of the algorithm of retransmission, we can determine the number of MTs in backlogged. The behaviour of this parameter is shown in Figure 4.
0.3
80
0.2
0.1
0.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Channel Load
Fig. 2 Behavior of the Throughput considering a Uniform Distribution and Capture Ratio.
Number of Backlogged Terminals
S - Normalised Throughput
In Figure 3, we can observe that the throughput improves when the capture effect (CE), the radio channel effects (RC) and stabilisation through the retransmission probabilities are used. We may say that this behaviour is because it uses a feedback channel which is useful for indicating the state of the channel, and thus controlling the re-transmission probabilities dynamically. With this the best performance of S-Aloha as RACH channel in the mobile cellular communication systems can be assured. Complying this way with the requirements established previously in the case of third generation mobile systems.
S-Aloha CE + RC + Stabilisation
70 60 50 40 30 20 10
We can observe in Figure 2 the influence of the capture ratio R in the behaviour of the throughput. For the case of almost perfect capture, R=2, the throughput presents the best response, reaching a 38% increase with respect to the theoretical S-Aloha. As the capture ratio increases the value of the throughput tends to decrease. When R=10 the throughput tends toward the S-Aloha values with an ideal channel (error-free channel). B. Behaviour of the Stability of S-Aloha The other parameter of interest in this work is to improve the stability of S-Aloha.; i. e., to increase the capacity of the system in the management of traffic. 0.8 0.7
Normalised Throughput
0.6 0.5
S-Aloha CE + RC+ Stabilisation
0.4 0.3 0.2
0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Normalised Throughput
Fig. 4 Number of mobile terminals in backlogged mode of S-Aloha, using an algorithm of retransmission adaptable, combined with the capture effect and radio channel effect. Figure 4 shows that the value of the MTs in backlogged for S-Aloha with ideal channel is high at maximum throughput. In our case, when applying the dynamic retransmission algorithm, the number of MTs in backlogged maintains a low value, less then 7% of the total of the mobile terminals. We could say upon applying a dynamic control on the retransmission probabilities through the use of a feedback channel, that it makes a more efficient use of the information channel, resolving the collisions more quickly, providing a decrease in the number of backlogged terminals in the region of low traffic. This has great significance facing another important parameter of S-Aloha as it is the propagation delay, which we discuss in the next section. D. Behaviour of the Average Delay of S-Aloha
0.1 0.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
Channel Load
Fig. 3 Behaviour of the Throughput improving the response of stability and efficiency.
According to the statistics obtained in Section III.C, regarding the number of MTs in backlogged and applying Little’s Theorem, [1], we can obtain the response of the average delay of S-Aloha.
S-Aloha CE + RC + Stabilisation
Average Delay (slots)
70 60 50 40 30 20 10 0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Normalised Throughput
Fig. 5 Behavior of the Average Delay of S-Aloha: typical response and enhanced response. According to Figure 5, we observe that when using the dynamic stabilisation algorithm, a minimum delay is present (almost zero) and with minimum variations until it reaches near 5 slots in the region of maximum throughput. Therefore, this behaviour is because the retransmission algorithm optimises the management of the channel, since it uses a dynamic control on the retransmission probabilities, resolving the collisions more quickly providing a decrease in the delay in the region of low traffic. We can conclude that the parameters which influence the performance of S-Aloha are, for throughput: the capture effect and the combined effect of the radio channel. In the case of the stability of the RACH channel, we can say that when using the ternary information provided in the feedback channel, the retransmission probability is set in an appropriate manner for the following slot. IV. CONCLUSIONS In this work, we have modelled and simulated a RACH channel with S-Aloha, considering a real environment. To increase the efficiency of the system, the capture effect was used with a uniform spatial distribution and the presence of the channel. Besides, an algorithm of dynamic retransmission was used with the feedback ternary information of the channel to control dynamically the load of the system. According to the results obtained in terms of the throughput, delay and terminals in backlogged state and stability, the latter is the one which influences more the performance of the system. We have found that as the mobile terminal knows the state of the channel in the previous slot, the number of collisions decreases, reflecting substantial improvements in such performance. We can conclude that to our belief, we highly recommend the use of the ternary information in the feedback channel. REFERENCES [1]
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